On the nonarchimedean quadratic Lagrange spectra

Markov and Lagrange Spectra for Laurent series in 1/T with rational coefficients

The field of formal Laurent series is a natural analogue of the real numbers, and mathematicians have been translating well-known results about rational approximations to that setting. In the framework of power series over the rational numbers, we define and study the Lagrange spectrum, related to Diophantine approximation of irrationals, and the Markov spectrum, related to representation by indefinite binary quadratic forms. We compute both spectra explicitly, and show that they coincide and exhibit no gaps, contrary to what happens over the reals.

The divided cell algorithm and the inhomogeneous Lagrange and Markoff spectra

The divided cell algorithm was introduced by Delone in 1947 to calculate the inhomogeneous minima of binary quadratic forms and developed further by E. S. Barnes and H. P. F. Swinnerton-Dyer in the 1950s. We show how advances of the past fifty years in both symbolic computation and our understanding of homogeneous spectra can be combined to make divided cells more useful for organizing information about inhomogeneous approximation problems. A crucial part of our analysis relies on work of Jane Pitman, who related the divided cell algorithm to the regular continued fraction algorithm. In partic…

Endomorphism rings of reductions of elliptic curves and abelian varieties

Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $Δ(v)$ of $End(E(v))$ is d…

Finiteness and periodicity of continued fractions over quadratic number fields

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Per…

扭曲线性模型的Epsilon二分性

Epsilon dichotomy for twisted linear models

本文研究特征零局部非阿基米德域二次扩张背景下，$\mathrm{GL}_n(D)$上扭曲线性模型的存在性与局部根数之间的Epsilon二分性关系。

Continued fractions for complex numbers and values of binary quadratic forms

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as also in a more general way from what we call "iteration sequences". We consider in this broader context the analogues of the Lagrange theorem characterizing quadratic surds, the growth properties of the denominators of the convergents, and the overall relation between sequences satisfying certain conditions, in terms of nonoccurrence of certain finite blocks, a…

关于二次无理数的高效逼近

On efficient approximation of quadratic irrationals, signed Chebyshev polynomials and Householder's method

本文提出高效计算二次无理数收敛的算法，证明在拉格朗日定理的伽罗瓦细化条件下，平方根的收敛序列的某些抽取是带符号的切比雪夫序列，并可通过Householder方法生成。

The Sum of Four Squares Over Real Quadratic Number Fields

Well-known results of Lagrange and Jacobi prove that the every $m \in \mathbb N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is $\mathbb Q(\sqrt{5})$. We provide explicit formulas for $r(m)$ for $K= \mathbb Q (\sqrt{2})$ and $K= \mathbb Q(\sqrt{5})$. We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bound…

Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set \[ {\mathcal L}(α)=\left\{\left.\limsup_{n\to \infty}\|ξα^n\|\ \right|\ ξ\in {\mathbb R}\right\}, \] where $\|x\|$ is the distance from $x$ to the nearest integer. First, we show that ${\mathcal L}(α)$ is closed in $[0,1/2]$ for any Pisot number $α$. Then we consider the case where $α$ is an integer with $α\geq 2$, or a quadratic unit with $α\ge 3$. We show that ${\mathcal …

部分商属于二次数域的连分数的周期性

On Periodicity of Continued fractions with Partial Quotients in Quadratic Number Fields

针对部分商取自实二次域整数环的连分数，本文研究其最终周期性与收敛性，给出实四次无理数对应周期展开的充要条件及构造算法。

正continued分数中Lagrange定理的几何证明

A geometric proof of Lagrange's theorem for continued fractions

本文通过几何方法证明了正continued分数中Lagrange定理，扩展了有理数和二次方程根的结论到更广泛的continued分数类型。

通过广义马尔可夫数的视角介绍拉格朗日谱与马尔可夫谱

An Introduction to the Lagrange and Markov Spectra through the Lens of Generalized Markov Numbers

本文综述拉格朗日谱与马尔可夫谱，通过广义马尔可夫数建立统一公式，连接经典理论并推广马尔可夫定理，提供系统证明与背景知识。

广义离散马尔可夫谱

Generalized discrete Markov spectra

本文提出广义马尔可夫理论，研究广义马尔可夫-拉格朗日谱，通过广义马尔可夫数扩展经典离散马尔可夫谱，利用蛇图替代Christoffel-word形式，构建广义马尔可夫方程谱，并证明其结构特性与贡献。